Indian Journal of Engineering & Materials Sciences Vol. 8, June 2001, pp. 123-128

Revisit to the stability of a uniform cantilever column subjected to Euler and Beck loads-Effect of realistic follower forces

G Venkateswara Rao & Gajbir Singh

Structural Engineering Group, Vikram Sarabhai Space Centre, Trivandrum 695 022, India

Received 25 August 2000; accepted 28 March 2001

The stability of a uniform cantilever column subjected to Euler and Beck loads is investigated for its practical significance and reported here. A classical solution to the system of equations of motion and constraints pertaining to the problem is presented in detail. The transcendental equation, so obtained, is solved using MAPLE-V software and interaction curves between Euler load and Beck load are plotted. The physical significance of these interaction curves is also discussed.

In his classic work, Euler l has shown that a column with a vertical load (conservative load) fails prematurely before it reaches the ultimate compressive load, because of instability. Since then, researchers in structural mechanics believed that a column subjected to a follower force (non­conservative load) does not fail because of instability. Beck2 showed in 1952 that such a column exhibits a dynamically unstable behaviour. Hence, the usage of terms like Euler instability or Beck instability and in other words Euler load or Beck load came into existence in literature. Accordingly, a cantilever column with a tip vertical compressive load is called Euler column and a cantilever column with tip tangential or follower force is called Beck column. The instabilities are further denoted as divergence when one of the frequencies becomes zero and flutter when two of the frequencies coalesce and subsequently become complex.

After Beck's work of stability of a cantilever column under follower compressive forces, a spate of research papers appeared in open literature around 1960s and 1970s and very good review articles on non-conservative stability of structural systems3

,4 and excellent books by Bolotin5, Leipholz6 and Ziegler? are known. Both continuum8.14 and finite element formulations I5-1? are published on this topic. Very accurate estimates of Beck load have been provided through the Galerkin finite element model of the problem I8-2o.

In a recent discussion, Koiter21 questioned the existence of follower forces and the experimental verification of the same and finally opined that papers on follower forces should not be accepted for publication in journals. However, Doak22 disagrees to

it suggesting an international debate on this issue, and saying that as long as the reviewers feel on the contrary, the articles on follower forces will be published.

In 1966, Willems23 proposed a method to experimentally simulate Beck load as a tip load at the free end of a cantilever column always passing through a fixed point. However, Huang et al. 24 have shown that even though this is a good attempt of experimental verification of Beck's critical load but does not exactly simulate Beck configuration and called it as Willems's configuration.

The present studies emphasize that the follower forces do exist and one need not be that pessimistic about the usefulness of studies made by various researchers on this topic. Based on our experience, we categorise the follower forces into the following categories, viz,: (i) The follower forces acting perpendicular to the beam axis are mainly due to aerodynamic forces acting on the wing of an aeroplane (lift forces) and its horizontal and vertical stabilizers25-28; (ii) The follower forces acting along the axis of the beam are mainly due to rocket thrust force29, skin friction drag force acting on the rockets30; and, (iii) In the case smart structures, it is possible to apply controlled follower force through sma.rt electro-mechanical/thermo-mechanicall magneto-mechanical actuators.

While the first type of follower forces (also called rotating loads) are studied by one of the present authors exhaustively25-28, follower forces of the second type were given as examples by Sugiyama et aZ.29 to represent Beck load or Leipholz load30 (a uniformly distributed follower force acting on the column). The third type of follower forces do occur in

124 INDIAN J. ENG. MATER. SCI. , JUNE 2001

smart structures application and can be used to our advantage and are discussed later.

A number of researchers have studied the effect of combined conservative loads and non-conservative loads acting on a column and predicted their stability behaviour3l

.34

.

Study of the behaviour of the column with tensile follower forces is very important to remove the misconception that internal pressure in a rocket motor (being a follower force by itself, can be idealized as two concentrated tensile follower forces at the ends of the rocket motor) has a stabilizing effect on the bending mode of instability which is of primary importance in missiles with very high ' g' loads. On the other hand, the concentrated tensile follower forces will actually destabilize the structure. This aspect is also discussed in this paper.

The problem considered is the same as considered by Celep31, viz prediction of the stability behaviour of a uniform cantilever column subjected to Euler and Beck loads at its free end. However, unlike Celep31,

who made a matter of fact statements about the types of instability, the real physical significance and practical importance of the study is brought out in the present study using user-friendly non-dimensional parameters and explanations.

Definition of the Problem Let us consider a uniform, isotropic cantilever

column, the undeformed and deformed positions being QR and QS respectively (Fig. 1).

Two loads, viz a vertical load Pe (Euler load -conservative) and Pb (Beck load - non-conservative) are applied simultaneously at the free end S of the column. It is required to find the interaction curves between Pb and Pe for the instability regions and find out whether the instability is of divergence or flutter

type. The non-dimensional parameters used are A~

(Beck critical load, defined as A~ = P b L2/rr,zEJ) and

A: (Euler critical load, defined as A: = 4 PeL2/rr2EJ).

Here, E is the Young's modulus, I is the moment of inertia and L is the length of the column. These non­dimensional parameters are chosen, specifically, for easy interpretation of the numerical results and to aITive at meaningful conclusions. In what follows, the formulation is briefly discussed and the method of solution of the governing equation of motion, followed in the present study is given.

Formulation of the Problem Following Timoshenko and Gere35 and Celep3 1, the

governing equation of motion of the column subjected to the loads Pe and Pb as shown in Fig. 1, is:

with boundary conditions

aw w=-=O at x=O ax and

... (1)

... (2)

(3)

The equation of motion given by Eq. (1) and boundary conditions represented by Eqs (2) & (3), can be written in the non-dimensional form as:

... (4)

x

L

y

Q y

Fig. i-Cantilever column subjected Euler and Beck loads

RAO & SINGH: REVISIT TO THE STABILITY OF A UNIFORM CANTILEVER COLUMN 125

... (5)

(6)

with the non-dimensional parameters defined as:

. . . (7)

The general solution of Eq. (4) can be assumed as:

w=( AcoshAI~ + BsinhAI~ + CCOSA2~ + DsinA2 ~ ) e i OJ r

where, w is the circular frequency .

... (8)

Substituting Eq. (8) into relation (4) and equating coefficients of hyperbolic and trigonometric terms leads to:

... (9)

and

.. . (10)

From equations (9) and (10), Af andA~ can be

written as:

_k2+~k4+40/a A2 ___ ~ __ --1- 2 .. . (11)

and

... (12)

Substituting Eq. (8) into the boundary condition Eqs (5) and (6), we get:

w(O)= A+ C=O ... (13)

. .. (14)

. .. (15)

... (16)

Eliminating A, B, C and D from relations (13)-(16), the final transcendental equation, which gives the solution for the present problem, is given by;

[(A~ + AIAe)sinhAl-(A~ - A2Ae)sinA2](Al sinhAl

+A2sin A2) - [(Af + Ae )coshAI+(A~ - Ae)COSA2 ]

(Af COShAl+A~COSA2 )=0 . . . (17)

Eq. (17) can be solved to obtain the eigen frequency aw2

, as described in the next section. It is interesting to note here that two degenerate

cases can be obtained by setting: (i) Ae=O and (ii) Ae=Ab=O (AI = ,1.2 = wv'a) . The first case corresponds to the classical Beck's column while the second case leads to the transcendental equation for free vibrations of a cantilever beam .

Case I: A.,,=O

k4 + 2 0/ a + kl Va sinh Al sinAl

+ 20/ a cosh Al COSAl=O

Case II : A.,,=4=0 0'1 = 1..2 = ro...Ja)

1 + cosh OJ ~ cos OJ ~ = 0

Method of Solution

... (18)

... (19)

Eq. (17) can be solved by using any standard numerical method to obtain the circular frequency awl for a given Ab and Ae or by using any standard package like MAPLE_y36. In the present study, we have used MAPLE-Y to obtain the solution. The solution procedure is:

Case I: Flutter instability For a given Ae, the value of Ab is increased from

zero to a value where the two frequencies coalesce.

126 INDIAN 1. ENG. MATER. SCI., JUNE 2001

This coalescence point gives a pair of critical values

that is,.1: and,.1~ (defined as, A: = 4 Pe L2 Ire 2 EI and

* 2/ ? ) ,.1b = Pb L re- EI

Case II: Divergence instability For a given ,.1b the values of ,.1e are increased from

zero to a value where one of the frequency becomes zero. This point once again gives a pair of critical

values A: and A~' Initially to validate the solution procedure, two

special cases of pure Beck load A: = 0 and pure Euler

10ad,.1~ = 0 are solved to obtain ,.1~ = 2.032 and

(J) Fa = 1l.011 for the former case and A: =l.0,

(J) Fa = 0.0 for the later case. It may be noted, these

are the exact classical results.

The values of A: and ,.1~ are plotted in Fig. 2 and

the behaviour of the interaction curves is discussed in the next section, highlighting the practical significance of the present study.

Results and Discussion Fig. 2 shows the interaction curves between A: and

,.1~. The curves are identified as various zones,

namely, BA, AD, CO, OA and EF. These zones have instability types as described by: (a) BA-First flutter; (b) eOA-First divergence; (c) AD-Second divergence; and, (d) EF-Third divergence. Out of these, the zones CO, OA, AB are of practical importance and are discussed in detail.

Zone CO

From zone CO it is clear that the A: decreases

rapidly as ,.1~ becomes negative. This implies that

when a cantilever column is subjected to a negative Beck load the Euler buckling load drastically decreases (destabilizing effect). In fact, a column

when subjected to tensile beck load ,.1~ = -6.0,

exhibits divergence type instability with A: "" O. This is contrary to the popular belief that for a long

cylinder (for example a rocket motor case with a fixity assumption to the previous stage) subjected to internal pressure the Euler buckling load increases. This technique is still being used to increase the so­called Euler stability load of upper stage motors of rockets. But as the internal pressure is a follower force which is equivalent to a tensile Beck load at the

ends of the motor case, and thus tends to destabilize the motor case subjected to Euler type of load due to structural assemblies sitting on it. It may be noted here that this phenomenon is applicable only for beam bending mode of buckling (column buckling only) and no conclusion can be made here for shell modes of buckling. As such, one has to be extra careful while making an attempt to stabilize the motor case by applying internal pressure.

Zone OA

It is evident from the interaction curve that A: in

the zone OA increases for a small positive value of

,.1~ . There is evidently a stabilizing effect on the

cantilever column when a small positive Beck load is acting on it. Please note that, in the present context, a positive Beck load is a compressive load. The increase in Euler critical load is qui te significant and this phenomenon can be effectively utilized using smart structures concept. The positive Beck load can be effectively applied by using a layer of piezo­ceramic (PZT) actuators (which induce a Beck type of load on the column). However, to bring out the usefulness of this phenomenon, experimental investigation of this study is absolutely necessary and the authors are at present working on that.

10,-----------------____________ ~

8

6

E 4 B / Flutter

2 ..c

*c< A

o · · ·······0· · ········ ·· ···· .... _--- ... ..... . -.. .. -- .- __ . __ . ... . F

D

-2

-4 Divergence

-6 c

o 4 8 16 20 24

Fig. 2-Interaction stability curves between criti cal Euler and Beck loads

RAO & SINGH: REVISIT TO THE STABILITY OF A UNIFORM CANTILEVER COLUMN 127

60 -0- A: b = 4.65 (E)

-0- A: b = 0.00 (F)

50 -6.- A: b = -0.56 (0)

-"1- A: b = 0.65 (A)

40

Fig. 3-Eigen curves for divergence at points A, D, E and F

Zone AB

It can be seen from Fig. 2 that critical Beck load is dependent on the sign and magnitude of the Euler load. When the Euler load is tensile, the critical Beck load increases with decrease in tensile Euler load. In case of compressive Euler load, the critical Beck load decreases with increase in the Euler load. If the Euler load is beyond point 'A', second divergence takes place (see interaction curve AD). It may be seen that third divergence instability shown by EF does not have any practical significance and is of theoretical interest only.

In Fig.3, the eigen curves at the points A, D, E and F are plotted. It is to be noted that the points D, E, F correspond to divergence instability points, where as, point A exhibit both divergence and flutter instability.

The values of A~ corresponding to these marked

points are given in the figure . The pairing values of

A: correspond to the intersection of these curves with

the x-axis (or A: axis).

Similarly, in Fig. 4, the eigen curves corresponding to marked points A and B are given for flutter instability. To trace these curves, data for one curve is

25

20 ..... 0 ' 0

15

10

"-......0

- 0 - ",'b = 0.65 (A

--0- ""9 = -2.00 (8

--""""""'0 ............... 0

~o

o~~~~~~~~~~~~~~~

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 . . A e or A b

Fig. 4-Coalescence curves corresponding to points A and B

obtained by fixing A: =-2.0 and varying Ab' The

coalescence of first two modes of free vibration takes

place at A: =-2.0 and A; =2.925. In the case of second

curve, A~ is fixed at 0.65 and Ae is varied to obtain

the coalescence point, which occurs at A~ =0.65 and

A: =4.095.

Conclusions The stability of a cantilever column subjected to

simultaneous Euler and Beck loads is investigated in this paper. The governing equations of motion and constraints are derived. These set of equations are solved using classical method and solution in the form of transcendental equation is obtained. The transcendental equation so obtained is solved using MAPLE-V. The results are presented in the form of user-friendly non-dimensional parameters. It is shown that tensile Beck load has a tendency to destabilize the column, unlike the popular belief otherwise. However, the stability margins can be increased by applying compressive follower forces, which can be easily attained by using smart structure concepts.

128 INDIAN 1. ENG. MATER. SCI., JUNE 2001

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